A Simple Path to Component Sizes in Critical Random Graphs

.We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs at criticality. Even though the main result is not new, we believe the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above-mentioned bounds.Keywordsrandom graphrandom walkballot theoremmartingaleMSC codes60G5005C8060C05